Introduction / Context:
This is a basic permutation question involving the arrangement of distinct objects in a row. Each subject book is considered different, and we are asked to find how many different linear arrangements are possible on a single shelf.
Given Data / Assumptions:
- Books: English, Hindi, Mathematics, History, Geography, Science.
- Total number of books = 6.
- All books are distinct.
- Order of books on the shelf matters.
Concept / Approach:
When arranging n distinct items in a row, the number of possible permutations is n factorial, written as n!. This is because we have n choices for the first position, n - 1 choices for the second position, and so on until the last position.
Step-by-Step Solution:
Step 1: Identify that there are 6 distinct books.
Step 2: The number of ways to arrange 6 distinct items in a row is 6!.
Step 3: Evaluate 6! = 6 * 5 * 4 * 3 * 2 * 1.
Step 4: Compute the product: 6 * 5 = 30, 30 * 4 = 120, 120 * 3 = 360, 360 * 2 = 720, and 720 * 1 = 720.
Step 5: Therefore, there are 720 distinct permutations.
Verification / Alternative check:
You can think sequentially: there are 6 choices for the first place, 5 remaining for the second place, 4 for the third, 3 for the fourth, 2 for the fifth and 1 for the last. Multiplying these together gives 6 * 5 * 4 * 3 * 2 * 1, which again is 720, confirming the factorial result.
Why Other Options Are Wrong:
360 equals 6! divided by 2 and would arise if some incorrect restriction or symmetry were applied. 240 and 780 do not correspond to any standard permutation count of 6 distinct items and result from miscalculations or wrong formula usage.
Common Pitfalls:
Some learners confuse permutations with combinations and try to use nCr instead of n!. Others may undercount by forgetting to include all positions in the factorial or overcomplicate a straightforward arrangement problem.
Final Answer:
The six different subject books can be arranged on the shelf in
720 distinct ways.
Discussion & Comments